An existence result for -Laplace equation with gradient nonlinearity in R

Abstract

We prove the existence of a weak solution to the problem −Δpu+V(x)|u|p−2uu(x)=f(u,|∇u|p−2∇u), >0 ∀x∈RN, where Δpu=div(|∇u|p−2∇u) is the p-Laplace operator, 1<p<N and the nonlinearity f:R×RN→R is continuous and it depends on gradient of the solution. We use an iterative technique based on the Mountain pass theorem to prove our existence result.